Why do test functions have to be real valued?

Question

Seems like if functions can be made to be complex-valued, they're going to be complex-valued. I have not seen anything about why test functions must be real valued.

Answer 1

There's a difference between interesting things that we want to study (e.g., distributions) and boring things we must have to make the machinery work (test functions). It's like having more resorts to fly to versus having more airport security checkpoints to pass.

If we introduce complex-valued test functions, the class of distributions does not change, but now we must perform more tests just to make sure the distribution we want to consider is well-defined (i.e., we must check it's continuous on complex-valued functions too). Why would you want to do extra work with no gain?

If we had complex-valued test functions, every other proof in distribution theory would begin with. "Let $\varphi$ be a test function. Write it as $\varphi = u+iv$ with $u,v$ real valued. We will consider $u$ below; the consideration of $v$ is similar..." Again, what for?

seems like if functions can be made to be complex-valued, they're going to be complex-valued.

Perhaps it only seems to you, while you are beginning to learn the subject. As you gain experience, you are likely to discover better reasons why things are studied in certain generality. For now, one word: spectrum.